The volume of simplices in high-dimensional Poisson-Delaunay tessellations

Abstract

Typical weighted random simplices $Z_{\mu }$, $\mu \in (-2,\infty )$, in a Poisson-Delaunay tessellation in ${\mathbb{R}}^n$ are considered, where the weight is given by the $(\mu +1)$st power of the volume. As special cases this includes the typical ($\mu =-1$) and the usual volume-weighted ($\mu =0$) Poisson-Delaunay simplex. By proving sharp bounds on cumulants it is shown that the logarithmic volume of $Z_{\mu }$ satisfies a central limit theorem in high dimensions, that is, as $n\to \infty$. In addition, rates of convergence are provided. In parallel, concentration inequalities as well as moderate deviations are studied. The set-up allows the weight $\mu =\mu (n)$ to depend on the dimension $n$ as well. A number of special cases are discussed separately. For fixed $\mu$ also mod-$\varphi$ convergence and the large deviations behaviour of the logarithmic volume of $Z_{\mu }$ are investigated.

References in corpus (6)

• A Central Limit Theorem for Convex Sets
• Limit theory for geometric statistics of point processes having fast decay of correlations
• Large deviations of the empirical volume fraction for stationary Poisson grain models
• Expected Sizes of Poisson-Delaunay Mosaics and Their Discrete Morse Functions
• A Central Limit Theorem for Vincular Permutation Patterns
• Random affine simplexes